Skip to main content
SpringerLink
Log in

Computational aspects of the mechanics of complex materials

  • Published:
Archives of Computational Methods in Engineering Aims and scope Submit manuscript

Summary

Bodies with exotic properties display material substructural complexity from nano to meso-level. Various models have been built up in condensed matter physics to represent the behavior of special classes of complex bodies. In general, they fall within the setting of an abstract model building framework which is not only a unifying structure of existing models but—above all—atool to construct special models of new exotic materials. We describe here basic elements of this framework, the one ofmultifield theories, trying to furnish a clear idea of the subtle theoretical and computational problems arising within it. We present the matter in a form that allows one to construct appropriate algorithms in special cases of physical interest. We discuss also issues related to the construction of compatible and mixed finite elements in linearized setting, the extension of extended finite element methods to analyze the influnce of material substructures on crack growth, the evolution of sharp discontinuity surfaces in complex bodies. Concrete examples of complex bodies are also presented with a number of details.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aero, E.L. and Kuvshinskii, E.V. (1960). Fundamental equations of the theory of elastic media with rotationally interacting particles.Fizika Tverdogo Tela,2, 1399–1409.

    Google Scholar 

  2. Antman, S.S. (1972). The theory of rods. inHandbuch der Physic Via/2.C. Truesdell ed., Springer Verlag, Berlin, 641–703.

    Google Scholar 

  3. Antman, S.S. (1995).Nonlinear problems of elasticity. Springer Verlag, Berlin, 1995.

    MATH  Google Scholar 

  4. Augusti, G. and Mariano, P.M. (1999). Stochastic evolution of microcracks in continua.Comp. Meth. Appl. Mech. Eng.,168, 155–171.

    MATH  MathSciNet  Google Scholar 

  5. Babuska, I. and Melenk, J.M. (1997). The partition of unity method.Int. J. Num. Meth. Eng.,40, 727–758.

    MATH  MathSciNet  Google Scholar 

  6. Bañas, L. and Slodička, M., (2005). Space discretization for the Landau-Lifshitz-Gilbert equation.Comp. Meth. Appl. Mech. Eng.,194, 467–477.

    MATH  Google Scholar 

  7. Belytschko, T. and Black, T. (1999). Elastic crack growth in finite elements with minimal remeshing.Int. J. Num. Meth. Eng.,45, 601–620.

    MATH  MathSciNet  Google Scholar 

  8. Bernardini, B. and Pence, T.J. (2004). A multifield theory for the modeling of the macroscopic behavior of shape memory materials. inAdvances in multifield theories for continua with substructure, G. Capriz and P.M. Mariano (Eds.), Birkhäuser, Boston, 199–242.

    Google Scholar 

  9. Biscari, P. (2002). Curvature effects on the surface viscosity of nematic liquid crystals.Eur. J. Mech. B Fluids 21, 739–750.

    MATH  MathSciNet  Google Scholar 

  10. Brezis, H. and Li, Y. (2001). Topology and Sobolev spaces.J. Funct. Anal.,183, 321–369.

    MATH  MathSciNet  Google Scholar 

  11. Brown, W.F. Jr. (1963).Micromagnetics, Wiley.

  12. Budiansky, B. and O’Connell, R.J. (1976). Elastic moduli of a cracked solid.Int. J. Solids Structures,12, 81–97.

    Google Scholar 

  13. Capriz, G. (1980). Bodies with microstructure. I..Riv. Mat. Univ. Parma,5, 673–691.

    MathSciNet  Google Scholar 

  14. Capriz, G., (1984). Continua with microstructures. (in Italian)Boll. Un. Mat. Ital. A,3, 181–195.

    MATH  MathSciNet  Google Scholar 

  15. Capriz, G. (1985). Continua with latent microstructure.Arch. Rational Mech. Anal.,90, 43–56.

    MATH  MathSciNet  Google Scholar 

  16. Capriz, G. (1985a). Introductory remarks to the dynamics of continua with microstructure. inMathematical models and methods in mechanics, 71–95, Banach Center Publ.,15, PWN, Warsaw.

    Google Scholar 

  17. Capriz, G. (1989).Continua with microstructure, Springer Verlag, Berlin.

    MATH  Google Scholar 

  18. Capriz, G. (1994). Smectic liquid crystals as continua with latent microstructure.Meccanica,30, 621–627.

    MathSciNet  Google Scholar 

  19. Capriz, G. (2003). Elementary preamble to a theory of granular gases.Rend. Sem. Mat. Univ. Padova,110, 179–198.

    MATH  MathSciNet  Google Scholar 

  20. Capriz, G. and Biscari, P. (1994). Special solutions in a generalized theory of nematics.Rend. Mat.,14, 291–307.

    MATH  MathSciNet  Google Scholar 

  21. Capriz, G. and Mariano, P.M. (2003). Symmetries and Hamiltonian formalism for complex materials.J. Elasticity,72, 57–70.

    MATH  MathSciNet  Google Scholar 

  22. Capriz, G. and Mariano, P.M. (2004). Balance at a junction among coherent interfaces in materials with substructure. inAdvances in multifield theories for continua with substructure, G. Capriz and P.M. Marino (Eds.), Birkhäuser, Boston, 243–263.

    Google Scholar 

  23. Capriz, G. and Mazzini, G. (1998). Invariance and balance in continuum mechanics. inNonlinear analysis and continuum mechanics (Ferrara, 1992), 27–35, Springer, New York.

    Google Scholar 

  24. Capriz, G. and Mullenger, G. (2004). Dynamics of granular fluids.Rend. Sem. Mat. Univ. Padova,111, 247–264.

    MATH  MathSciNet  Google Scholar 

  25. Capriz, G. and Podio-Guidugli, P. (1976). Discrete and continuous bodies with affine structure.Ann. Math. Pura Appl.,115, 195–217.

    MathSciNet  Google Scholar 

  26. Capriz, G. and Podio-Guidugli, P. (1977). Formal structure and classification of theories of oriented materials.Ann. Math. Pura Appl.,111, 17–39.

    MathSciNet  Google Scholar 

  27. Capriz, G. and Podio-Guidugli, P. (1981). Materials with spherical structure.Arch. Rational Mech. Anal.,75, 269–279.

    MATH  MathSciNet  Google Scholar 

  28. Capriz, G. and Podio-Guidugli, P. (1983). Structured continua from a Lagrangian point of view.Ann. Mat. Pura Appl.,135, 1–25.

    MATH  MathSciNet  Google Scholar 

  29. Capriz, G., Podio-Guidugli, P. and Williams, W. (1982). On balance equations for materials with affine structure.Meccanica,17, 80–84.

    MATH  Google Scholar 

  30. Capriz, G. and Virga, E.G. (1990). Interactions in general continua with microstructure.Arch. Rational Mech. Anal.,109, 323–342.

    MATH  MathSciNet  Google Scholar 

  31. Capriz, G. and Virga, E.G. (1994). On singular surfaces in the dynamics of continua with microstructure.Quart. Appl. Math.,52, 509–517.

    MATH  MathSciNet  Google Scholar 

  32. Carstensen, C. and Praetorius, D. (2005). Effective simulation of a macroscopic model for stationary micromagnetics.Comp. Meth. Appl. Mech. Eng.,194, 531–548.

    MATH  MathSciNet  Google Scholar 

  33. Chessa, J. and Belytschko, T. (2003). On construction of blending elements for locally enriched methods.Int. J. Num. Meth. Eng.,57, 1015–1038.

    MATH  MathSciNet  Google Scholar 

  34. Chirita, S. and Aron, M. (1999). Aspects of Saint-Venant’s principle in the dynamical theory of linear micropolar elasticity.Math. Mech. Solids,4, 17–34.

    MATH  MathSciNet  Google Scholar 

  35. Choksi, R., Kohn, R.V. and Otto, F. (2004). Energy minimization and flux domain structure in the intermediate state of a type-I superconductor.J. Nonlinear Sci.,14, 119–171.

    MATH  MathSciNet  Google Scholar 

  36. Ciarlet, P. (1988).Mathematical elasticity, Vol. I. Three-dimensional elasticity, North-Holland Publishing Co., Amsterdam.

    Google Scholar 

  37. Ciarlet, P.G. (1991).Basic error estimates for elliptic problems. Handbook of numerical analysis, Vol. II. 17–351. North-Holland. Amsterdam.

    Google Scholar 

  38. Coleman, B.D. and Gurtin, M.E. (1967). Thermodynamics with internal state variables.J. Chem. Phys.,47, 597–613.

    Google Scholar 

  39. Coleman, B.D. and Noll, W. (1963). The thermodynamics of elastic materials with heat conduction and viscosity.Arch. Rational Mech. Anal.,13, 245–261.

    MATH  MathSciNet  Google Scholar 

  40. Coleman, B.D. and Owen, D.R. (1974). A mathematical foundation for thermodynamics.Arch. Rational Mech. Anal.,54, 1–104.

    MATH  MathSciNet  Google Scholar 

  41. Colli, P., Frémond, M. and Visintin, A. (1990). Thermo-mechanical evolution of shape memory alloys.Quart. Appl. Math.,48, 31–47.

    MATH  MathSciNet  Google Scholar 

  42. Cosserat E. and Cosserat F. (1909).Sur la theorie des corps deformables. Dunod, Paris.

    Google Scholar 

  43. Cowin, S.C. (1985). The viscoelastic behavior of linear elastic materials with voids.J. Elasticity,15, 185–191.

    MATH  Google Scholar 

  44. Cowin, S.C. and Nunziato, J.W. (1983). Linear elastic materials with voids.J. Elasticity,13, 125–147.

    MATH  Google Scholar 

  45. Davì, F. (2001). On domain switching in deformable ferroelectrics, seen as continua with microstructure.Z. Angew. Math. Phys.,52, 966–989.

    MATH  MathSciNet  Google Scholar 

  46. Davì, F. and Mariano, P.M. (2001). Evolution of domain walls in ferroelectric solids.J. Mech. Phys. Solids,49, 1701–1726.

    MATH  Google Scholar 

  47. Davì, F. and Rizzoni, R. (2004). On twinning and domain switching in ferroelectric Pb(Zr1-x Tix)O3. I. Twins and domain walls.J. Mech. Phys. Solids,52, 113–144.

    MATH  MathSciNet  Google Scholar 

  48. De Angelis, E., Casciola, C.M., Mariano, P.M. and Piva, R. (2004). Microstructure and turbulence in dilute polymer solutions. inAdvances in multifield theories for continua with substructure, G. Capriz and P. M. Mariano (Eds.), Birkhäuser, Boston, 127–148.

    Google Scholar 

  49. de Fabriitis, C. and Mariano, P.M. (2005). Geometry of interactions in complex bodies.J. Geom. Phys.,54, 301–323.

    MathSciNet  Google Scholar 

  50. De Gennes, P.-G. and Prost, J. (1993).The physics of liquid crystals, Oxford University Press, Oxford.

    Google Scholar 

  51. De Simone, A., Kohn, R.V., Müller, S. and Otto, F. (2002). A reduced theory for thin-film micromagnetics.Comm. Pure Appl. Math.,55, 1408–1460.

    MathSciNet  Google Scholar 

  52. Del Piero, G. (2003). A class of fit regions and a universe of shapes for continuum mechanics.J. Elasticity,70, 175–195.

    MATH  MathSciNet  Google Scholar 

  53. Del Piero, G. and Owen, R.D. (2000).Structured deformations. Quaderni dell’Istituto di Alta Matematica, Florence.

    Google Scholar 

  54. Dietsche, A., Steinmann, P. and Willam, K. (1993). Micropolar elasto-plasticity and its role in localization analysis.Int. J. Plasticity,9, 813–831.

    MATH  Google Scholar 

  55. Dolbow, J., Moës N. and Belytschko T. (2001). An extended finite element method for modeling crack growth with frictional contact.Comp. Meth. Appl. Mech. Eng.,190, 6825–6846.

    MATH  Google Scholar 

  56. Doyle, T.C. and Ericksen, J.L. (1956). Nonlinear elasticity.Adv. Appl. Mech.,4, 53–115.

    MathSciNet  Google Scholar 

  57. Dunn, J.E. and Serrin, J. (1985). On the thermomechanics of interstitial working.Arch. Rational Mech. Anal.,88, 95–133.

    MATH  MathSciNet  Google Scholar 

  58. E, W. (1997). Nonlinear continuum theory of smectic-A liquid crystals.Arch. Rational Mech. Anal.,137, 159–175.

    MATH  MathSciNet  Google Scholar 

  59. Epstein, M. and de Leon, M. (1998). Geometrical theory of uniform Cosserat media.J. Geom. Phys.,26, 127–170.

    MATH  MathSciNet  Google Scholar 

  60. Ericksen, J.L. (1960). Theory of anisotropic fluids.Trans. Soc. Rheol.,4, 29–39.

    MathSciNet  Google Scholar 

  61. Ericksen, J.L. (1961). Conservation laws for liquid crystals.Trans. Soc. Rheol.,5, 23–34.

    MathSciNet  Google Scholar 

  62. Ericksen, J.L. (1962a). Kinematics of macromolecules.Arch. Rational Mech. Anal.,9, 1–8.

    MathSciNet  Google Scholar 

  63. Ericksen, J.L. (1962b). Hydrostatic theory of liquid crystals.Arch. Rational Mech. Anal.,9, 371–378.

    MATH  MathSciNet  Google Scholar 

  64. Ericksen, J.L. (1977). Special topics in elastostacis.Adv. Appl. Mech.,17, 189–244.

    MATH  Google Scholar 

  65. Ericksen, J.L. (1991). Liquid crystals with variable degree of orientation.Arch. Rational Mech. Anal.,113, 97–120.

    MATH  MathSciNet  Google Scholar 

  66. Ericksen, J.L. and Truesdell, C.A. (1958). Exact theory of stress and strain in rods and shells.Arch. Rational Mech. Anal.,1, 295–323.

    MATH  MathSciNet  Google Scholar 

  67. Eringen, C.A. (1973). On nonlocal microfluid mechanics.Int. J. Eng. Sci.,11, 291–306.

    MathSciNet  Google Scholar 

  68. Eringen, A.C. (Ed.) (1975).Continuum physics. Vol II. Continuum Mechanics of single-substance bodies. Academic Press, New York-London.

    Google Scholar 

  69. Eringen, A.C. (1999).Microcontinuum field theories. I. Foundations and solids. Springer-Verlag, New York.

    MATH  Google Scholar 

  70. Fetecau, R.C., Marsden, J.E., Ortiz, M. and West, M. (2003). Nonsmooth Lagrangian mechanics and variational collision integrators.SIAM J. Dyn. Syst.,2, 381–416.

    MATH  MathSciNet  Google Scholar 

  71. Fleming, M., Chu, Y.A., Moran, B. and Belytschko, T. (1997). Enriched element-free galerkin methods for crack tip fields.Int. J. Num. Meth. Eng.,40, 1483–1504.

    MathSciNet  Google Scholar 

  72. Fox, D.D. and Simo, J.C. (1992). A nonlinear geometrically exact shell model incorporating independent (drill) rotations.Comp. Meth. Appl. Mech. Eng.,98, 329–343.

    MATH  MathSciNet  Google Scholar 

  73. Frémond, M. (1987). Matériaux à mémoire de forme.C. R. Acad. Sci. Paris,304, 239–244.

    Google Scholar 

  74. Frémond, M. (2000).Non-smooth thermomechanics. Springer-Verlag, Berlin.

    Google Scholar 

  75. Frémond, M. and Nedjar, B. (1996). Damage, gradient of damage and principle of virtual power.Int. J. Solids Structures,33, 1083–1103.

    MATH  Google Scholar 

  76. Fulton, C.C. and Gao, H. (2001). Microstructural modeling of ferroelectric fracture.Acta Materialia,49, 2039–2054.

    Google Scholar 

  77. García-Cervera, C.J., Gimbutas, Z. and Gimbutas, E. W. (2003). Accurate numerical methods for micromagnetics simulations with general geometries.J. Comput. Phys.,184, 37–52.

    MATH  MathSciNet  Google Scholar 

  78. Germain, P. (1973). The method of virtual power in continuum mechanics, Part 2: microstructure.SIAM J. Appl. Math.,25, 556–575.

    MathSciNet  Google Scholar 

  79. Goodman, M.A. and Cowin, S.C. (1972). A continuum theory of granular materials.Arch. Rational Mech. Anal.,44, 249–266.

    MATH  MathSciNet  Google Scholar 

  80. Gordon, A. (1986). Interface motions in ferroelectrics.Physica B,138, 239–243.

    Google Scholar 

  81. Gordon, A. (1991). Propagation of solitary stress waves at first-order ferroelectric phase transitions.Phys. Lett. A,154, 79–80.

    Google Scholar 

  82. Gordon, A. (2001). Finite-size effects in dynamics of paraelectric-ferroelectric interfaces induced by latent heat transfer.Phys. Lett. A,281, 357–362.

    MATH  Google Scholar 

  83. Gravouil, A., Moës, N. and Belytschko, T. (2002). Non-planar 3D crack growth by the extended finite element and level sets Part II: Level set update.Int. J. Num. Meth. Eng.,53, 2569–2586.

    Google Scholar 

  84. Green, A.E. and Laws, N. (1966). A general theory of rods.Proc. Roy. Soc. Lond. A,293, 145–155.

    Google Scholar 

  85. Green, A.E. and Naghdi, P.M. (1995). A unified procedure for construction of theories of deformable media. II. Generalized Continua.Proc. Royal Soc. London A,448, 357–377.

    MathSciNet  Google Scholar 

  86. Green, A.E., Naghdi, P.M. and Wainwright, W.L. (1965). A general theory of a Cosserat surface.Arch. Rational Mech. Anal.,20, 287–308.

    MathSciNet  Google Scholar 

  87. Grioli, G. (1960). Elasticità asimmetrica.Ann. Mat. Pura Appl.,50, 389–417.

    MATH  MathSciNet  Google Scholar 

  88. Grioli, G. (2003). Microstructures as a refinement of Cauchy theory. Problems of physical concreteness.Cont. Mech. Thermodyn.,15, 441–450.

    MATH  MathSciNet  Google Scholar 

  89. Gurtin, M.E. (1965). Thermodynamics and the possibility of spatial interactions in elastic materials.Arch. Rational Mech. Anal.,19, 339–352.

    MATH  MathSciNet  Google Scholar 

  90. Gurtin, M.E. (1995). The nature of configurational forces.Arch. Rational Mech. Anal.,131, 67–100.

    MATH  MathSciNet  Google Scholar 

  91. Gurtin, M.E. (1996). Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance.Physica D,92, 178–192.

    MATH  MathSciNet  Google Scholar 

  92. Gurtin, M.E. (1997). Dynamical theories of electromagnetism and superconductivity based on gauge invariance and energy.Arch. Rational Mech. Anal.,137, 49–97.

    MATH  MathSciNet  Google Scholar 

  93. Gurtin, M.E. and Struthers, A. (1990). Multiphase thermomechanics with interfacial structure. III. Evolving phase boundaries in the presence of bulk deformation.Arch. Rational Mech. Anal.,112, 97–160.

    MATH  MathSciNet  Google Scholar 

  94. Hall, E.O. (1951). The deformation and aging of mild steel.Proc. Phys. Soc. B,64, 747–753.

    Google Scholar 

  95. Hashin, Z. (1988). The differential scheme and its applications to cracked materials.J. Mech. Phys. Solids,36, 719–734.

    MATH  MathSciNet  Google Scholar 

  96. Huang, Y., Hu, K.X. and Chandra, A. (1994). A generalized self-consistent mechanics method for microcracked solids.J. Mech. Phys. Solids,42, 1273–1291.

    MATH  Google Scholar 

  97. Holm, D.D. (2002). Euler-Poincaré dynamics of perfect complex fluids. inGeometry, Mechanics and Dynamics, 113–167, Springer, New York.

    Google Scholar 

  98. Hu, C., Wang, R. and Ding, D.-H. (2000). Symmetry groups, physical property tensors, elasticity and dislocations in quasicrystals.Rep. Prog. Phys.,63, 1–39.

    MathSciNet  Google Scholar 

  99. Iesan, D. (1987). Saint-Venant’s problem.Lecture Notes in Mathematics, Springer-Verlag, Berlin.

    MATH  Google Scholar 

  100. Iesan, D. (2002). On the theory of heat conduction in micromorphic continua.Internat. J. Engrg. Sci.,40, 1859–1878.

    MathSciNet  Google Scholar 

  101. James, R.D. and Rizzoni, R. (2000). Pressurized shape memory thin films.J. Elasticity,59, 399–436.

    MATH  MathSciNet  Google Scholar 

  102. Krajcinovic, D. (1996).Damage mechanics, North-Holland, Amsterdam.

    Google Scholar 

  103. Krajcinovic, D. and van Mier, J. (Eds.) (2000).Damage and fracture in disordered materials. Springer, Wien.

    Google Scholar 

  104. Landau, L.D. and Lifshitz, E.M. (1968).Course of theoretical physics. Vol. 5: Statistical physics. Pergamon Press, Oxford-Edinburgh-New York.

    Google Scholar 

  105. Leslie, F.M. (1968). Some constitutive equations for liquid crystals.Arch. Rational Mech. Anal.,28, 265–28.

    MATH  MathSciNet  Google Scholar 

  106. Lew, A., Marsden, J.E., Ortiz, M. and West, M. (2003). Asynchronous variational integrators.Arch. Rational Mech. Anal.,167, 85–146.

    MATH  MathSciNet  Google Scholar 

  107. Likos, C.N. (2001). Effective interactions in soft condensed matter physics.Phys. Rep.,348, 267–439.

    Google Scholar 

  108. Mariano, P.M. (1995). Fracture in structured continua.Int. J. Damage Mech.,4, 283–289.

    Google Scholar 

  109. Mariano, P.M. (1999). Some remarks on the variational decription of microcracked bodies.Int. J. Non-Linear Mech.,34, 633–642.

    MATH  Google Scholar 

  110. Mariano, P.M. (2000). Configurational forces in continua with microstructure.Z. angew. Math. Phys. ZAMP,51, 752–791.

    MATH  MathSciNet  Google Scholar 

  111. Mariano, P.M. (2000). Premises to a multified approach to stochastic damage evolution. InDamage and Fracture in Disordered Materials, D. Krajcinovic and J. van Mier (Eds.), Springer-Verlag, Berlin, 217–263.

    Google Scholar 

  112. Mariano, P.M. (2001). Coherent interfaces with junctions in continua with microstructure.Int. J. Solids Structures,38, 1243–1267.

    MATH  MathSciNet  Google Scholar 

  113. Mariano, P.M. (2001). Multifield theories in mechanics of solids.Adv. Appl. Mech.,38, 1–93.

    MathSciNet  Google Scholar 

  114. Mariano, P.M. (2002). A note on Ceradini-Capurso-Maier’s theorem in plasticity.Int. J. Plasticity,18, 1749–1773.

    MATH  Google Scholar 

  115. Mariano, P.M. (2002). Cancellation of vorticity in steady-state non-isentropic flows of complex fluids.J. Phys. A: Math. Gen.,36, 9961–9972.

    MathSciNet  Google Scholar 

  116. Mariano, P.M. (2005). Influence of material substructure on crack propagation: a unified treatment.Proc. Royal Soc. London A,461, 371–395.

    MathSciNet  Google Scholar 

  117. Mariano, P.M. (2005). Migration of substructures in complex fluids.L. Phys. A,38, 6823–6839.

    MATH  MathSciNet  Google Scholar 

  118. Mariano, P.M. (2005). Mechanics of quasiperiodic alloys.J. Nonlinear Sci., in print.

  119. Mariano, P.M. (2005).Elements of multified theories of complex bodies. Birkhauser, Boston, (in preparation).

    Google Scholar 

  120. Mariano, P.M. and Augusti, G. (1998). Multifield description of microcracked continua. A local model.Math. Mech. Solids,3, 237–254.

    MathSciNet  Google Scholar 

  121. Mariano, P.M. and Bernardini, D. (1998). Flow rules for porous elastic-plastic materials.Mech. Res. Comm.,25, 443–448.

    MATH  MathSciNet  Google Scholar 

  122. Mariano, P.M., Casciola, C.M. and De Angelis, E. (2004). Substructural interactions and transport in polymer flows.Int. J. Non-Linear Mech.,39, 457–465.

    MATH  Google Scholar 

  123. Mariano, P.M., Gioffrè, M., Stazi, F.L. and Augusti, G. (2004). Elastic microcracked bodies with random properties.Prob. Eng. Mech.,19, 127–143.

    Google Scholar 

  124. Mariano, P.M. and Stazi, F.L. (2001). Strain localization in elastic microcracked bodies.Comp. Meth. Appl. Mech. Eng.,190, 5657–5677.

    MATH  Google Scholar 

  125. Mariano, P.M. and Stazi, F.L. (2004). Strain localization due to crack-microcrack interactions: X-FEM for a multifield approach.Comp. Meth. Appl. Mech. Eng.,193, 5035–5062.

    MATH  Google Scholar 

  126. Mariano, P.M., Stazi, F.L. and Augusti, G. (2004). Phason effects around a crack in Al−Pb−Mn quasicrystals: stochastic aspects of the phonon-phason coupling.Comp. Stru.,82, 971–983.

    Google Scholar 

  127. Mariano, P.M. and Trovalusci, P. (1999). Constitutive relations for elastic microcracked bodies: from a lattice model to a multifield continuum description,Int. J. of Damage Mech.,8, 153–173.

    Google Scholar 

  128. Markov, K.Z. (1995). On a microstructural model of damage in solids,Int. J. Engng Sci.,33, 139–150.

    MATH  Google Scholar 

  129. Marsden, J.E. and Hughes, T.J.R. (1994).Mathematical foundations of elasticity, Prentice Hall, Dover edition.

  130. Marsden, J.E. and West, M. (2001). Discrete mechanics and variational integrators,Acta Numerica,10, 357–514.

    MATH  MathSciNet  Google Scholar 

  131. McLean, D. (1987). Dislocation contribution to the flow stress of polycristalline iron,Can. J. Phys.,45, 973–982.

    Google Scholar 

  132. Mermin, N.D. (1979). The topological theory of defects in ordered media,Rev. Mod. Physics,51, 591–648.

    MathSciNet  Google Scholar 

  133. Michel, L. (1980). Symmetry defects and brocken symmetry configuration hidden symmetry,Rev. Mod. Phys.,52, 617–651.

    Google Scholar 

  134. Mielke, A. (2002). Analysis of energetic models for rate-independent materials,Proc. Int. Congress Mathematicians, Vol III, Higher Ed. Press, Beijing, 817–828.

    Google Scholar 

  135. Mielke, A. (2004). Deriving new evolution equations for microstructures via relaxation of variational incremental problems,Comp. Meth. Appl. Mech. Eng.,193, 5095–5127.

    MATH  MathSciNet  Google Scholar 

  136. Mielke, A., Theil, F. and Levitas, V.I. (2002). A variational formulation of rate-independent phase transformations using an extremum principle,Arch. Rational Mech. Anal.,162, 137–177.

    MATH  MathSciNet  Google Scholar 

  137. Mindlin, R.D. (1964). Micro-structure in linear elasticity.Arch. Rational Mech. Anal.,16, 51–78.

    MATH  MathSciNet  Google Scholar 

  138. Moës, N., Gravouil, A. and Belytschko, T. (2001). Non-planar 3D crack growth by the extended finite element and level sets Part I: Mechanical model,Int. J. Num. Meth. Eng.,53, 2549–2568.

    Google Scholar 

  139. Moran, B. and Shih, C.F. (1987). A general treatment of crack tip contour integrals,Int. J. Fracture,35, 295–310.

    Google Scholar 

  140. Mori, T. and Tanaka, K. (1973). Average stress in matrix and average elastic energy of materials with misfitting inclusions,Acta Metall.,21, 571–574.

    Google Scholar 

  141. Mosconi, M. (2002). Mixed variational formulations for continua with microstructure,Int. J. Solids Structures,39, 4181–4195.

    MATH  MathSciNet  Google Scholar 

  142. Musesti, A. (2004). The balance equations of continuum mechanics in geometric measure theory, (in Italian),Boll. Unione Mat. Ital. Sez. B,7, 305–317.

    MATH  MathSciNet  Google Scholar 

  143. Nash, J.F. (1954). C1 isometric imbeddings,Ann. Math.,60, 383–396.

    MathSciNet  Google Scholar 

  144. Nash, J.F. (1956). The imbedding problem for Riemannian manifold,Ann. Math.,63, 20–63.

    MathSciNet  Google Scholar 

  145. Nix, W.D. (1988). Mechanical properties of thin filins,Metall. Trans. A,20, 2217–2245.

    Google Scholar 

  146. Noll, W. (1973). Lectures on the foundations of continuum mechanics and thermodynamics,Arch. Rational Mech. Anal.,52, 62–69.

    MATH  MathSciNet  Google Scholar 

  147. Nowaki, W. (1986).Theory of asymmetric elasticity, Pergamon Press, New York.

    Google Scholar 

  148. Nunziato, J.W. and Cowin, S.C. (1979). A nonlinear theory of elastic materials with voids,Arch. Rational Mech. Anal.,72, 175–201.

    MATH  MathSciNet  Google Scholar 

  149. Nunziato, J.W. and Walsh, E.K. (1978). On the influence of void compaction and material non-uniformity on the propagation of one-dimensional acceleration waves in granular materials,Arch. Rational Mech. Anal.,64, 299–316.

    MathSciNet  Google Scholar 

  150. Pedregal, P. (1993). Laminates and microstructure,European J. Appl. Math.,4, 121–149.

    MATH  MathSciNet  Google Scholar 

  151. Penrose, O. and Fife, P.C. (1990). Thermodynamically consistent models of phase field type for the kinetics of phase transitions,Physica D,43, 44–62.

    MATH  MathSciNet  Google Scholar 

  152. Petch, N.J. (1953). The cleavage strength of polycrystals,J. Iron Steel Inst. London,174, 25–28.

    Google Scholar 

  153. Povstenko, Y.Z. (1994). Stress functions for continua with couple stresses,J. Elasticity,36, 99–116.

    MATH  MathSciNet  Google Scholar 

  154. Qin, Q.-H., Mai, Y.-W. and Yu, S.-W. (1998). Effective moduli for thermopiezoelectric materials with microcracks,Int. J. Fracture,91, 359–371.

    Google Scholar 

  155. Rubin, M.B. and Benveniste, Y. (2000). A Cosserat shell model for interphases in elastic media,J. Mech. Phys. Solids,52, 1023–1052.

    MathSciNet  Google Scholar 

  156. Segev, R. (1994). A geometrical framework for the static of materials with microstructure,Math. Mod. Meth. Appl. Sci.,4, 871–897.

    MATH  MathSciNet  Google Scholar 

  157. Segev, R. (2004). Fluxes and flux-conjugated stresses, inAdvances in Multified Theories of Continua with Substructure, G. Capriz and P.M. Mariano (Ed.), Birkhäuser, Boston, 149–163.

    Google Scholar 

  158. Shu, Y.C. and Bhattacharya, K. (2001). Domain patterns and macroscopic behavior of ferroelectric materials,Phil Mag. B,81, 2021–2054.

    Google Scholar 

  159. Shu, J.Y. and Fleck, N.A. (1999). Strain gradient crystal plasticity: size dependent deformation of bicrystals,J. Mech. Phys. Solids,47, 297–324.

    MATH  Google Scholar 

  160. Šilhavý, M. (1978). A condition equivalent to the existence of non-equilibrium entropy and temperature for materials with internal variables.Arch. Rational Mech. Anal.,68, 299–332.

    MathSciNet  Google Scholar 

  161. Šilhavý, M. (1985). Phase transitions in non-simple bodies,Arch. Rational Mech. Anal.,88, 135–161.

    Google Scholar 

  162. Šilhavý, M. (1991). Cauchy’s stress theorem and tensor fields with divergences in Lp,Arch. Rational Mech. Anal.,116, 223–255.

    MathSciNet  Google Scholar 

  163. Šilhavý, M. (1997).The mechanics and thermodynamics of continuous media, Springer Verlag, Berlin, 1997.

    Google Scholar 

  164. Simo, J.C. and Fox, D.D. (1989). On a stress resultant geometrically exact shell model. Part I: Formulation and optimal parametrization,Comp. Meth. Appl. Mech. Eng.,72, 267–304.

    MATH  MathSciNet  Google Scholar 

  165. Simo, J.C., Fox, D.D. and Hughes, T.J.R. (1992). Formulations of finite elasticity with independent rotations,Comp. Meth. Appl. Mech. Eng.,95, 277–288.

    MATH  MathSciNet  Google Scholar 

  166. Simo, J.C., Fox, D.D. and Rifai, M.S. (1989). On a stress resultant geometrically exact shell model II. The linear theory: computational aspects,Comp. Meth. Appl. Mech. Eng.,73, 53–92.

    MATH  MathSciNet  Google Scholar 

  167. Simo, J.C., Fox, D.D. and Rifai, M.S. (1990). On a stress resultant geometrically exact shell model III. Computational aspect of the nonlinear theory,Comp. Meth. Appl. Mech. Eng.,79, 21–70.

    MATH  MathSciNet  Google Scholar 

  168. Simo, J.C., Marsden, J.E. and Krishnaprasad, P.S. (1988). The Hamiltonian structure of nonlinear elasticity: the material and convective representations of solids, rods and plates.Arch. Rational mech. Anal.,104, 125–183.

    MATH  MathSciNet  Google Scholar 

  169. Stackgold, I. (1950). The Cauchy relations in a molecular theory of elasticity.Quart. Appl. Math.,8, 169–186.

    MathSciNet  Google Scholar 

  170. Stazi, F.L. (2003). Finite element methods for cracked and microcracked bodies,Ph.D. Thesis, Universitá di Roma “La Sapienza”, Roma (Italy) (avalaible athttp://www.cimne.com/eccomas/html/awd1.htm).

    Google Scholar 

  171. Stazi, F.L., (2004). Consequences of inavariance under changes of observers in a multifield model of microcracked bodies,Meccanica,39, 389–393.

    MATH  Google Scholar 

  172. Stazi, F.L., (2004). Finite element methods for cracked and microcracked bodies,Proceedings of the European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2004), Jyväskylä, 24–28 July.

  173. Stazi, F.L., Budyn, E., Chessa, J. and Belytschko, T. (2002). An Extended Finite Elment Method with Higher-Order Element for Crack Problems with Curvature.Computational Mechanics,31, 38–48.

    Google Scholar 

  174. Steinmann, P. (1994). A micropolar theory of finite deformation and finite rotation multiplicative elastoplasticity.Int. J. Solids Str.,31, 1063–1084.

    MATH  MathSciNet  Google Scholar 

  175. Stewart, J.R. and Hughes, T.J.R. (1998). A tutorial in elementary finite element error analysis: a systematic presentation of a priori and a posteriori error estimates.Comp. Meth. Appl. Mech. Eng.,158 (1998), 1–22.

    MATH  MathSciNet  Google Scholar 

  176. Stolarska, M., Chopp, D.L., Moës, N. and Belytschko, T. (2001). modelling crack growth by level sets and the extended finite element method.Int. J. Num. Meth. Eng.,51, 943–960.

    MATH  Google Scholar 

  177. Sukumar, N., Moës, N., Moran, B. and Belytschko, T. (2000). Extended finite element method for three-dimensional crack modeling,Int. J. Num. Meth. Eng.,48 (11), 1549–1570.

    MATH  Google Scholar 

  178. Tiersten, H.F. (1964). Coupled magnetomechanical equations for magnetically saturated insulators,J. Math. Phys.,5, 1298–1318.

    MATH  MathSciNet  Google Scholar 

  179. Truesdell, C.A. and Noll, W. (2004).The non-linear field theories of mechanics. Third edition. Springer-Verlag, Berlin.

    Google Scholar 

  180. Truesdell, C.A. and Toupin, R.A. (1960). Classical field theories of mechanics, inHandbuch der Physics, Springer Verlag, Berlin.

    Google Scholar 

  181. Voigt, W. (1887). Studien über die Elasticitätsverhältuisse der Krystalle,Abh. Ges. Wiss. Göttingen,34.

  182. Voigt, W. (1894). Über Medien ohne innere Kräfte und ein durch gelieferte mechanische Dentung der Maxwell-Hertzschen Gleichungen,Gött. Abh., 72–79.

  183. Wang, X.-P., García-Cervera, C.J. and García-Cervera, E. W. (2001). A Gauss-Seidel projection method for micromagnetics simulations,J. Comput. Phy.,171, 357–372.

    MATH  Google Scholar 

  184. Wei, Y. and Hutchinson, J.W. (1997). Steady-state crack growth and work of fracture for solids characterized by strain gradient plasticity.J. Mech. Phys. Solids,45, 1253–1273.

    MATH  MathSciNet  Google Scholar 

  185. Whitney, H. (1936). Differentiable manifolds,Ann. Math.,37, 645–680.

    MathSciNet  Google Scholar 

  186. Zhang, W. and Bhattacharya, K. (2005). A computational model of ferroelectric domains. Part I: model formulation and domain switching,Acta Materalia,53, 185–198.

    Google Scholar 

  187. Zhang, W. and Bhattacharya, K. (2005). A computational model of ferroelectric domains. Part I: model formulation and domain switching.Acta Materalia,53, 199–209.

    Google Scholar 

  188. Zi, G. and Belytschko, T. (2003). New crack-tip elements for XFEM and applications to cohesive cracks.Int. J. Num. Meth. Eng.,57, 2221–2240.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Università di Firenze, Firenze, Italy

    Paolo Maria Mariano

  2. Università di Roma “La Sapienza”, Roma, Italy

    Furio Lorenzo Stazi

Authors
  1. Paolo Maria Mariano
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Furio Lorenzo Stazi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Paolo Maria Mariano.

Additional information

Dedicated to Gianfranco Capriz on occasion of his eightieth birthday

Rights and permissions

Reprints and permissions

About this article

Cite this article

Mariano, P.M., Stazi, F.L. Computational aspects of the mechanics of complex materials. ARCO 12, 391–478 (2005). https://doi.org/10.1007/BF02736191

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02736191

Keywords

Search

Navigation